ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Shabunin A. V., Nikolaev S. M., Astakhov V. V. Two-parametric bifurcational analysis of regimes of complete synchronization in ensemble of three discrete-time oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 6, pp. 24-39. DOI: 10.18500/0869-6632-2005-13-5-24-39

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Language: 
Russian
Article type: 
Article
UDC: 
517.9

Two-parametric bifurcational analysis of regimes of complete synchronization in ensemble of three discrete-time oscillators

Autors: 
Shabunin Aleksej Vladimirovich, Saratov State University
Nikolaev Sergej Mihajlovich, Saratov State University
Astakhov Vladimir Vladimirovich, Yuri Gagarin State Technical University of Saratov
Abstract: 

We invetsigate mechanisms of appearance and disappearance of regimes of complete synchronization of chaos in a ring of three logistic maps with symmetric diffusive coupling. Two-parametric bifurcational analysis is carried out and typical oscillating regimes and transitions between them are considered.

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Reference: 
  1. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator system. Progress of Theoretical Physics. 1983;69(1):32–47. DOI: 10.1143/PTP.69.32.
  2. Pikovsky AS. On the interaction of strange attractors. Preprint of the Institute of Applied Physics of the Academy of Sciences of the USSR. Gorky; 1983. 20 p. (in Russian).
  3. Kuznetsov SP. Universality and scaling in the behavior of coupled Feigenbaum systems. Radiophys. Quantum Electron. 1985;28(8):681–695. DOI: 10.1007/BF01035195.
  4. Afraimovich VS, Verichev NN, Rabinovich MI. Stochastic synchronization of oscillation in dissipative systems. Radiophys. Quantum Electron. 1986;29(9):795–803. DOI: 10.1007/BF01034476.
  5. Hasler M, Maistrenko Y, Popovych O. Simple example of partial synchronization of chaotic systems. Phys. Rev E. 1998;58(5):6843–6846. DOI: 10.1103/PhysRevE.58.6843.
  6. Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HDI. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E. 1995;51(2):980–994. DOI: 10.1103/physreve.51.980.
  7. Abarbanel HDI, Rulkov NF, Sushchik MM. Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E. 1996;53(5):4528–4535. DOI: 10.1103/PhysRevE.53.4528.
  8. Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Forced and mutual synchronization of chaos. Sov. J. Commun. Technol. Electron. 1991;36(2):338–351 (in Russian).
  9. Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Synchronization of chaos. Int. J. Bifurcat. Chaos. 1992;2(3):633–644. DOI: 10.1142/S0218127492000756.
  10. Rosenblum MG, Pikovsky AS, Kurths J. Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 1996;76(11):1804–1807. DOI: 10.1103/physrevlett.76.1804.
  11. Belykh VN, Mosekilde E. One-dimensional map lattices: synchronization, bifurcations, and chaotic structures. Phys. Rev. E. 1996;54(4):3196–3203. DOI: 10.1103/PhysRevE.54.3196.
  12. Brown R, Rulkov NF. Synchronization of chaotic systems: transverse stability of trajectories in invariant manifolds. Chaos. 1997;7(3):395–413. DOI: 10.1063/1.166213.
  13. Andreyev YV, Dmitriev AS. Conditions for global synchronization in lattices of chaotic elements with local connections. Int. J. Bifurcat. Chaos. 1999;9(11):2165–2172. DOI: 10.1142/S0218127499001589.
  14. Astakhov V, Shabunin A, Kapitaniak T, Anishchenko V. Loss of chaos synchronization through the sequence of bifurcations of saddle periodic orbits. Phys. Rev. Lett. 1997;79(6):1014–1017. DOI: 10.1103/PhysRevLett.79.1014.
  15. Astakhov V, Hasler M, Kapitaniak T, Shabunin A, Anishchenko V. Effect of parameter mismatch on the mechanism of chaos synchronization loss in coupled systems. Phys. Rev. E. 1998;58(5):5620–5628. DOI: 10.1103/PhysRevE.58.5620.
  16. Maistrenko Y, Maistrenko V, Popovych O, Mosekilde E. Desynchronization of chaos in coupled logistic maps. Phys. Rev. E. 1999;60(3):2817–2830. DOI: 10.1103/PhysRevE.60.2817.
  17. Astakhov V, Shabunin A, Klimshin A, Anishchenko V. In-phase and antiphase complete chaotic synchronization in symmetrically coupled discrete maps. Discrete Dynamics in Nature and Society. 2000;7(4):215–229. DOI: 10.1155/S1026022602000250.
  18. Astakhov V, Shabunin A, Uhm W, Kim S. Multistability formation and synchronization loss in coupled Henon maps: Two sides of the single bifurcational mechanism. Phys. Rev. E. 2001;63(5):056212. DOI: 10.1103/PhysRevE.63.056212.
  19. Ashwin P, Buescu J, Stewart I. Bubbling of attractors and synchronization of chaotic oscillators. Phys. Lett. A. 1994;193(2):126–139. DOI: 10.1016/0375-9601(94)90947-4.
  20. Ashwin P, Buescu J, Stewart I. From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity. 1996;9(3):703–738. DOI: 10.1088/0951-7715/9/3/006.
  21. Venkataramani SC, Hunt BR, Ott E. Bubbling transition. Phys. Rev. E. 1996;54(2):1346–1360. DOI: 10.1103/PhysRevE.54.1346.
  22. Taborov AV, Maistrenko YL, Mosekilde E. Partial synchronization in a system of coupled logistic maps. Int. J. Bifurcat. Chaos. 2000;10(5):1051–1066. DOI: 10.1142/S0218127400000748.
Received: 
15.07.2005
Accepted: 
17.10.2005
Published: 
28.02.2006
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